Subgroup ($H$) information
| Description: | $D_{21}$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(946\)\(\medspace = 2 \cdot 11 \cdot 43 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
32 & 9 \\
34 & 19
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
25 & 34 \\
17 & 18
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
29 & 21 \\
22 & 13
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $\PSL(2,43)$ |
| Order: | \(39732\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
| Exponent: | \(19866\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,43)$, of order \(79464\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
| $\operatorname{Aut}(H)$ | $S_3\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $W$ | $D_{21}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $D_{21}$ | ||
| Normal closure: | $\PSL(2,43)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\PSL(2,43)$ | ||
| Maximal under-subgroups: | $C_{21}$ | $D_7$ | $S_3$ |
Other information
| Number of subgroups in this conjugacy class | $946$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(2,43)$ |